Spectrally constrained impulse shortening filter for a discrete multi-tone receiver

ABSTRACT

A channel in a multiple carrier communication system is equalized by computing a target spectral response, shortening the impulse response of the channel so that a significant part of an energy of the impulse response is confined to a region that is shorter than a target length and filtering the signal based on the target spectral response.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. Ser. No. 09/054,468, filedApr. 3, 1998 now U.S. Pat. No. 6,631,175.

BACKGROUND

The invention relates to time-domain equalization in a discretemulti-tone (DMT) receiver.

Conventional single carrier modulation techniques translate data bitsfor transmission through a communication channel by varying theamplitude and/or phase of a single sinusoidal carrier. By contrast, DMT,which is also referred to as Orthogonal Frequency Division Multiplexing(OFDM) or Multicarrier Modulation (MCM), employs a large number ofsinusoidal subcarriers, e.g., 128 or 256 subcarriers. The availablebandwidth of the communication channel is divided into subchannels andeach subchannel communicates a part of the data. A DMT system may employquadrature amplitude modulation (QAM) for each of the subcarriers.

OFDM-based systems transmit blocks of information bits. The timerequired to transmit one such block is called the symbol period. Thetime domain waveform that corresponds to one such block of bits iscalled a symbol.

Intersymbol interference (ISI) arises from the characteristics ofpractical communication channels and limits the rate at whichinformation can be transmitted through them. Specifically, communicationchannels typically have an Effective Discrete-Time Impulse Response(EDIR) that is greater than one sample time in length, which causes ISI.ISI is a well-known phenomenon in single-carrier communication systemsand there are many techniques for reducing it. The process of such ISIreduction is called equalization. ISI is discussed, for example, inProakis, Digital Communications, McGraw Hill, 2nd Edition, 1989.

Equalization in OFDM-based Systems is achieved by a two stage process.First, at the transmitter, an end-portion of each symbol is affixed tothe beginning of the symbol to form what is called a Cyclic Prefix (CP).A cyclic prefix that is greater than the EDIR of the channel preventsone symbol from interfering with another. Furthermore, it alsofacilitates a simple method of neutralizing the time-domain spread ofeach symbol forced by the channel. This is achieved by the use of asimple frequency domain process in the receiver which requires onemultiplication operation for each used subcarrier of the OFDM system.

The use of a Cyclic Prefix to reduce ISI is discussed, for example, in:Cimini, “Analysis and Simulation of a Digital Mobile Channel usingOrthogonal Frequency Division Multiplexing,” IEEE Transactions oncommunications, pp 665-675 July 1985; Chow, “A Discrete Multi-ToneTransceiver System for HDSL applications,” IEEE Journal on SelectedAreas of Communications, 9(6):895-908, August 1991; “DMT Group VDSL PMDDraft Standard Proposal,” Technical Report, T1E1.4/96-329R2, ANSI 1997.

Another problem arising in conventional DMT systems is noise bleeding,which is the phenomenon of noise in one frequency band interfering witha signal whose subcarrier is in another frequency band. Noise bleedingis caused, in general, by the Discrete Fourier Transform (DFT) operationat the receiver. Noise bleeding is discussed in, for example, Worthenet. al., “Simulation of VDSL Test Loops,” Technical ReportT1E1.4/97-288, ANSI 1997.

In a perfectly synchronized DMT system, the signal in one frequency banddoes not interfere with a signal whose subcarrier is in anotherfrequency band. However, noise from one band may interfere with otherless noisy bands and render them unusable. Techniques for dealing withnoise-bleeding include wavelet-based solutions. However, wavelet-basedsolutions are, in general, computationally intensive.

SUMMARY

The invention provides a spectrally constrained impulse shorteningfilter (SCISF) for use, for example, in DMT systems. The SCISF servestwo primary functions.

First, the SCISF reduces intersymbol interference (ISI) by reducing thelength of the effective discrete-time impulse response (EDIR) of thecommunication channel. Conventional impulse shortening filters may havedeep nulls in their frequency response. By contrast, the SCISF has afilter characteristic that is essentially free from undesired nulls thatmay attenuate or completely eliminate certain subcarriers.

Second, the SCISF reduces noise bleeding between subchannels byattenuating noisy channels in a manner that does not reduce the signalto noise ratio (SNR) in these channels, but reduces the noise power thatmay appear in the sidelobes of adjacent subchannels. The SCISFaccomplishes these functions by applying a frequency constraint to thesignal based on a target spectral response.

In one general aspect, the invention features equalizing a channel in amultiple carrier communication system. The system includes a spectrallyconstrained impulse shortening filter. Received noise power spectraldensity is measured. A target spectral response is computed based on themeasured noise power. A frequency response for the spectrallyconstrained impulse shortening filter is selected based on the targetspectral response. The communication signal is filtered with thespectrally constrained impulse shortening filter.

Embodiments may include one or more of the following features. The noisepower spectral density may be measured at the output of the discreteFourier transform. The spectrally constrained impulse shortening filtermay be a time domain digital filter.

In another aspect, the invention features equalizing a channel in amultiple carrier communication system. The channel has an impulseresponse and is configured to receive a signal having a cyclic prefix. Atarget spectral response is computed. The impulse response of thechannel is shortened so that a significant part of the energy of theimpulse response is confined to a region that is shorter than a targetlength, and the signal is filtered based on the target spectralresponse.

The target length may be a length of the cyclic prefix. The targetspectral response may be computed from measured noise power density,which may be measured at the output of a discrete Fourier transform. Forexample, the target spectral response may be the inverse of the measurednoise power spectral density. The filtering step may be performed with afilter having a frequency response selected to match the target spectralresponse. The shortening of the impulse response and/or the filteringmay be performed by a time domain digital filter.

In another general aspect, the invention features selecting an impulseresponse for a spectrally constrained impulse shortening filter in amultiple carrier communication system. Received noise power spectraldensity is measured. A cost function is computed using the noise power.The cost function is dependent on the impulse response. Thedimensionality of a space over which the cost function is defined isreduced and the cost function is minimized. The noise power spectraldensity may be measured at the output of a discrete Fourier transform.The cost function may be used to compute coefficients for the spectrallyconstrained impulse shortening filter.

In another general aspect, the invention features a spectrallyconstrained impulse shortening filter for a multiple carriercommunication system. The system includes a channel that has an impulseresponse. A digital filter structure receives the signal and apply afrequency characteristic to the signal. The frequency characteristic isdetermined by filter coefficients. Filter coefficients are selected toshorten the impulse response of the channel so that a significant partof the energy of the impulse response is confined to a region that isshorter than a target length and to apply a frequency characteristic tothe signal based on a target spectral response. The selected filtercoefficients are input to the taps of the filter.

In another general aspect, the invention features a receiver forreceiving a multiple carrier signal from a communication channel havingan impulse response. An analog-to-digital converter receives the signalfrom the communication channel. A spectrally constrained impulseshortening filter receives the signal from the analog-to-digitalconverter and shortens the impulse response of the channel so that asignificant part of an energy of the impulse response is confined to aregion that is shorter than a target length. The filter also applies afrequency characteristic to the signal based on a target spectralresponse. A discrete Fourier transform receives the output of thespectrally constrained impulse shortening filter and a decoder receivesoutputs of the discrete Fourier transform.

In another aspect, the invention features a modem in which an encoderreceives digital data and outputs a constellation of complex values. Aninverse discrete Fourier transform receives the constellation from theencoder. A digital-to-analog converter is connected to the inversediscrete Fourier transform and outputs a signal to a communicationchannel. An analog-to-digital converter receives the signal from thecommunication channel. A spectrally constrained impulse shorteningfilter shortens an impulse response of the channel so that a significantpart of an energy of the impulse response is confined to a region thatis shorter than a target length. The signal is filtered based on atarget spectral response. A discrete Fourier transform connected to thefilter and a decoder is connected to the discrete Fourier transform andoutputs digital data.

In another aspect, the invention features software for causing aprocessor in a communication system to perform the following operations:measure received noise power spectral density; and compute a targetspectral response based on the measured noise power. The software mayalso include instructions for causing a processor to compute filtercoefficients based on the target spectral response.

In another aspect, the invention features software for causing aprocessor in a communication system to measure received noise powerspectral density. A cost function is computed using the noise power. Thecost function is dependent on an impulse response of a spectrallyconstrained impulse shortening filter. The dimensionality of a spaceover which the cost function is defined is reduced, and the costfunction is minimized.

The techniques described here are not limited to any particular hardwareor software configuration. They may find applicability in any computingor processing environment that may be used for a communication system.The techniques may be implemented in hardware or software, or acombination of the two. Preferably, the techniques are implemented incomputer programs executing on a digital signal processor that includesa processor and a storage medium readable by the processor (includingvolatile and non-volatile memory).

Other features and advantages will be apparent from the followingdescription, including the drawings and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a discrete multi-tone communication system.

FIG. 2 is a plot of effective discrete-time impulse response (EDIR) of acommunication channel including transmit and receive filters.

FIG. 3 is a plot of the shortened EDIR due to a spectrally constrainedimpulse shortening filter (SCISF).

FIG. 4 is a block diagram of a SCISF.

FIG. 5 is a plot of transmit signal power, signal power at SCISF inputand noise power at SCISF input.

FIG. 6 is a plot of signal and noise power at the output of the SCISF.

FIG. 7 shows the filter response of a discrete Fourier transform for onefrequency band.

FIG. 8 is a plot of the target spectral response of the SCISF, G_(d)(ω),versus the actual frequency response, G(ω).

FIG. 9 is a plot of signal-to-noise ratio at the output of the SCISF,the output of the DFT with the SCISF and the output of the DFT withoutthe SCISF.

DESCRIPTION

As shown in FIG. 1, a discrete multi-tone (DMT) communication system 10has a transmitter 12 and a receiver 14. The transmitter 12 accepts aninput data bit stream which passes through a constellation encoder 20.The encoder 20 divides the serial input bit stream into blocks of data.These blocks of data are further subdivided into smaller blockscorresponding to subchannels. Each of these smaller blocks are used tocompute a complex value representing a constellation point. Eachconstellation point corresponds to a subsymbol. The subsymbols are thenoutput by the encoder. Taken together, the subsymbols constitute asymbol.

The subsymbols are supplied to an inverse discrete Fourier transform(IDFT) 30, which may be implemented, for example, in a digital signalprocessor. The IDFT 30 outputs N time samples of a symbol. The timesamples are processed by a parallel to serial converter 40 to form asingle stream of time samples.

Following the parallel to serial converter 40, a prefixer 50 adds acyclic prefix to the beginning of each symbol to reduce intersymbolinterference (ISI). Alternatively, the cyclic prefix may be added in theparallel to serial converter. After the cyclic prefix is added, theresulting signal passes through a digital-to-analog (D/A) converter 60for transmission to the receiver 14 through a communication channel 70.An analog transmit filter 65 may be included following the D/A converterto band limit the transmitted signal.

At the receiver 14, the signal passes through an analog-to-digital (A/D)converter 80 and then through an impulse shortening filter 90. An analogreceive filter 75 may be included prior to the A/D converter in order toband limit the received signal. A prefix stripper 100 strips the cyclicprefixes from the resulting symbols and a serial to parallel converter110 divides the stream of time samples into parallel signal paths thatform the inputs to a discrete Fourier transform (DFT) 120. The DFT 120converts the time samples into subsymbols. A decoder 130 converts thesubsymbols into a data bits and outputs the resulting data.

As discussed above, a cyclic prefix is added to each symbol prior totransmission through the communication channel to reduce the effects ofISI. The cyclic prefix is formed by copying the last ν time samples fromthe end of a symbol and placing them at the beginning of the symbol. Toeliminate ISI, the length of the cyclic prefix, ν, is chosen to belonger than the effective discrete-time impulse response (EDIR) of thechannel. However, because the cyclic prefix constitutes redundant data,increasing the length of the cyclic prefix reduces the efficiency of thecommunication system. For example, in a system having N time samples persymbol and a cyclic prefix of ν time samples, the efficiency of thesystem will be reduced by N/(N+ν). To maximize efficiency it isnecessary either to minimize ν or to maximize N. However, increasing Nincreases the complexity, latency and computational requirements of thesystem and at some point becomes impractical. Accordingly, it isdesirable to minimize ν.

An impulse shortening filter having an impulse response, g(n), may beemployed in the receiver to minimize the length of the cyclic prefix bydecreasing the EDIR, g(n)*h(n), of the effective communication channel,which includes the transmit and receive filters, the impulse shorteningfilter and the physical transmission channel. The use of an impulseshortening filter is referred to as time domain equalization. Decreasingthe EDIR allows a shorter cyclic prefix to be used without increasingISI.

FIG. 2 is a plot of the EDIR for a DMT test configuration having acommunication channel that is 4500 feet in length and operates at asampling frequency of 11.04 MHz (test loop 4, as described in “Very-highSpeed Digital Subscriber Lines: System Requirements,” Technical ReportT1E1.4/97-131R1, ANSI 1998). The EDIR includes the effects of a transmitfilter, the communication channel and a receive filter. FIG. 3 shows theimpulse response as shortened by the addition of an impulse shorteningfilter.

The impulse shortening filter is selected so that a significant part ofthe energy of the joint impulse response of the filter and the effectivecommunication channel, g(n)*h(n), is confined to a region that isshorter in length than the length of the cyclic prefix. Some prioralgorithms for computing g(n) considered only shortening the EDIR anddid not consider the spectral characteristics of the resulting impulseshortening filter. Such filters often had deep nulls in some frequencybands, which rendered some of the corresponding subchannels useless.

Since increasing the length of the cyclic prefix reduces systemefficiency, the receiver may dynamically compute an optimal length forthe cyclic prefix and may send that information to the transmitter. Forexample, the receiver may compute a set of impulse responses for theimpulse shortening filter based on a set of predetermined cyclic prefixlengths. The receiver then computes the system throughput for eachparticular cyclic prefix length. The length that maximizes systemthroughput is selected and the result of that selection is communicatedto the transmitter. The transmitter then operates using the selectedcyclic prefix length.

To avoid the possible attenuation of frequency bands, the spectralresponse of the impulse shortening filter is further required to have aspectral response, G(ω), that meets a specified spectral constraint. Aspectral constraint of the form G(ω)H(ω)>τ, where τ is a threshold, issufficient to avoid nulls in the frequency response of the impulseshortening filter. However, it is possible to compute a spectralconstraint or target spectral response, Gd(ω), that provides additionalperformance improvements, such as reducing noise bleeding betweensubchannels. A spectrally constrained impulse filter is configured tohave a spectral response that approximates the target spectral response.

As shown in FIG. 4, the spectrally constrained impulse shortening filter(SCISF) 90 may be implemented as a time domain digital filter, which hasa digital filter structure 210 with a number of taps 220 or filtercoefficient inputs for adjusting the filter response. The coefficientsmay be computed and supplied to the taps by a digital signal processor(DSP). Alternatively, the SCISF may be implemented entirely in software,i.e., within a DSP.

A desired spectral response may be applied to a received signal using afilter that is separate from the impulse shortening or time domainequalization (TEQ) filter. For example, an analog filter may be placedprior to the A/D converter. However, the adjustability of such a filterwould be limited. As a further example, a digital filter could be addedprior to the TEQ filter. Both of these configurations suffer thedisadvantage that the TEQ filter may distort the desired spectralcharacteristics of the added filter. A filter also might be positionedafter the TEQ filter, which would reduce noise bleeding, but mightreduce the impulse shortening provided by the TEQ filter. Accordingly,the SCISF integrates the TEQ (i.e., impulse shortening) function withthe desired spectral response in a single filter.

In summary, the filter characteristic, g(n), of the SCISF satisfies twoconditions. First, the effective length of the convolution of the filtercharacteristic with the impulse response of the communication channel,g(n)*h(n), is less than a target length. Second, the cost function(error function) between the target spectral response, G_(d)(ω), and theactual filter spectral response, G(ω), is minimized.

The target spectral response is an ideal filter characteristic that isselected to maximize the data bit throughput in the subchannels of a DMTsystem by reducing the impact of noise. There are many sources for noisein a DMT communication system, such as near-end cross-talk (NEXT), radiofrequency interference (RFI) and noise generated in the communicationchannel (white noise). As shown in FIGS. 5 and 6, the noise spectraldensity is generally not uniform across the frequency band of thecommunication system. This non-uniformity contributes to the problem ofnoise bleeding, in which noise in one frequency band interfering with asignal in another frequency band.

In general, noise bleeding is caused by sidelobes of filters in a DFT.FIG. 7 shows the filter response of a DFT for one frequency band or bin(i.e., bin 128). The first sidelobes (250) are only 13 dB below the mainlobe (260). Therefore, noise located outside of bin 128, but within thefirst sidelobe of bin 128, i.e., approximately mid-way between bins 126and 127, would appear in bin 128 with an attenuation of only 13 dB.Consequently, noisy subchannels in a DMT system may degrade theperformance of non-noisy subchannels.

The target spectral response is essentially a spectral constraint thatattenuates noisy channels more than non-noisy channels. The signal andnoise in the noisy channels are reduced equally, so the attenuation doesnot affect the signal-to-noise ratio in these channels. However, becausethe absolute noise level in the noisy channels is reduced, there is lessnoise available to enter the sidelobes of the non-noisy channels. Hence,the noise bleeding problem is minimized.

To determine the target spectral response, the noise power spectraldensity (noise PSD) at the receiver must be known. The noise PSD may bedetermined, for example, by performing a periodigram on received data.This measurement is more complicated if a transmitter is transmitting,since the transmitted signal must be separated from the noisemeasurement. The noise PSD is determined by: (i) slicing the receivedconstellation of subcarriers after the DFT to determine the nearestvalid constellation point; (ii) determining an error signal based on thedifference between the received constellation point and the validconstellation point; (iii) performing an IDFT on the error signal; and(iv) generating a periodigram (with windowing) from the error signals.The noise PSD may then be determined from the periodigram.

An example of a noise PSD characteristic for a DMT communication systemis shown in FIG. 5 (test loop 4, as described in “Very-high SpeedDigital Subscriber Lines: System Requirements,” Technical ReportT1E1.4/97-131R1, ANSI 1998). The transmit signal power is measured atthe output of the transmit filter in the transmitter. The signal andnoise PSD plots shown in FIG. 5 are measured at the input of the A/Dconverter in the receiver, which is prior to the SCISF.

Measured noise PSD is used by a digital signal processor (DSP) tocompute a target spectral response, G_(d)(ω), using the algorithmdescribed below. Alternatively, the inverse of the noise PSD may be usedas an approximation for the target spectral response. A spectralresponse, G(ω), is then determined for the SCISF that minimizes theerror between the spectral response of the SCISF and the target spectralresponse. A set of filter coefficients may then be generated toconfigure the SCISF to the determined characteristic. These calculationsmay be done periodically to adjust the performance of the communicationsystem. Frequency domain equalization coefficients and symbolsynchronization may also be adjusted based on these calculations.

FIG. 8 is a plot of the target spectral response of the SCISF, G_(d)(ω),versus the actual frequency response, G(ω). The difference between theresponses is only a few dB. FIG. 6 shows the signal and noise PSD at theoutput of the SCISF.

FIG. 9 shows the dramatic effect of the SCISF on system performance.Without the SCISF (i.e., using a filter that provides only impulseshortening), the signal-to-noise ratio (SNR) decreases significantly atthe output of the Fourier transform (i.e., FFT or DFT) to less thanabout 7 dB. This decrease is due, in large part, to noise bleedingcaused by the sidelobes of the Fourier transform. By contrast, with theSCISF, the SNR at the output of the Fourier transform tracks the SNR atthe output of the SCISF within a few dB. Overall, the SCISF provides animprovement in SNR.

The target spectral response of the SCISF, G_(d)(ω) and the actualfrequency response, G(ω), are derived from an energy constraint for theSCISF, i.e., the target spectral response must localize the energy ofthe effective impulse response within a desired frequency band. Theenergy constraint is combined with a desired spectral response based onthe measured noise power spectral density. The resulting cost function(or error function) is then minimized to obtain a practical filtercharacteristic for the SCISF. This process is presented in detail below.

In the following derivation, all vectors are column vectors by default.Vectors are denoted by bold lower case letters (e.g., t). The size m_(t)of a vector t, is written t(m_(t)). The components of a vector aredenoted by lower case letters, e.g., t(m_(t))=[t₀ . . . t_(m) _(t)⁻¹]^(T). The convolution of length m_(t)+m_(h)−1 of vectors t(m_(t)) andh(m_(h)) is denoted t*h. The Toeplitz matrix of a vectort(m_(t)),t_((m:n)) Δ[t_(m) . . . t_(n)]^(T), is written

T_((m : n))^(m_(x)), and  T_((m : n))^(m_(x))x = (t * x)_((m : n)).The Discrete Time Fourier Transform (DTFT) of a vector t may berepresented as:

$\begin{matrix}{{T(\omega)} = {\sum\limits_{n = 0}^{m_{t} - 1}{t_{n}{{\mathbb{e}}^{{- j}\;\omega\; n}.}}}} & (1)\end{matrix}$

The EDIR of the channel and the impulse response of the SCISF areexpressed as h(m_(h)) and g(m_(g)), respectively. The cyclic prefix hasa length of m_(c) per symbol. The energy of the effective impulseresponse must be localized to a contiguous region of target lengthm_(l), where m_(l)<m_(c), while satisfying a spectral constraint. Theenergy criterion to be satisfied by the SCISF may be written as:

$\begin{matrix}{\frac{{\left( {g*h} \right)_{({m:{m + m_{l} - 1}})}}^{2}}{{\left( {g*h} \right)}^{2}} \geq \alpha} & (2)\end{matrix}$for some 0≦m<m_(g)+m_(h)−2 and some 0<α<1. Defining the set S^(m) as

$\begin{matrix}{s^{m} = \left\{ {g \in {R^{m_{g}}:{\frac{{\left( {g*h} \right)_{({m:{m + m_{l} - 1}})}}^{2}}{{\left( {g*h} \right)}^{2}} \geq \alpha}}} \right\}} & (3)\end{matrix}$the impulse response of the SCISF must belong to S^(m) for some m.

Let ω_(l), . . . , ω_(N) be the location of the sub-carriers in thefrequency domain. The spectral constraint may be applied by selectinggεS^(m), so that the cost function

$\begin{matrix}{{J(g)}\overset{\Delta}{=}{\sum\limits_{i = 1}^{N}\left( {{{G\left( \omega_{i} \right)}} - {G_{d}\left( \omega_{i} \right)}} \right)^{2}}} & (4)\end{matrix}$is minimized for the desired target spectral response G_(d)(ω).

Typically, this optimization would have to be repeated for all possiblem to select the filter impulse response, g, that achieves the lowestpossible value of J. However, as discussed below, optimization may belimited to a few well chosen values of m.

The determination of the filter impulse response may be done in twostages. First, the desired magnitude frequency response G_(d)(ω) of theimpulse shortening filter g over the bins used by the DMT system isobtained. Second, J is optimized over S^(m) for a specific value of m.

To determine the target spectral response G_(d)(ω), it is necessary touse expressions for the signal to noise ratios observed in the variousfrequency bins at the output of the DFT in the receiver. The DMT systemhas M tones (subcarriers), N of which (those from M₁ through M₂) areused, and the communication channel has an analog frequency responseH_(c)(ƒ). The analog noise power spectral density observed at the inputof the receiver A/D is S_(η)(ƒ). Prior to conversion in the A/Dconverter, the received analog signal may be filtered by ananti-aliasing filter with transfer function H_(α)(ƒ). The EDIR in theabsence of the impulse shortening filter is h(n). After the A/Dconverter, the signal is fed into the impulse shortening filter with animpulse response of g(n). The impulse shortening filter ensures that(h(n)*g(n)) is shorter in length than the cyclic prefix. G(ω) is thediscrete time Fourier transform of g(n). Under these conditions, theexpected signal energy μ(k) observed in bin k at the output of thelength-2M receiver DFT is given by:

$\begin{matrix}{{{\mu(k)} = {C_{1}D_{k}{{H\left( \frac{\pi\; k}{M} \right)}}^{2}{{G\left( \frac{\pi\; k}{M} \right)}}^{2}}};{{H(\omega)} = {{H_{c}\left( \frac{\omega}{2\pi\; T} \right)}{H_{a}\left( \frac{\omega}{2\pi\; T} \right)}}}} & (5)\end{matrix}$where C₁ is a constant, 1/T the sampling frequency and D_(k) thetransmitted power in bin k. The noise power η(k) in bin k is:

$\begin{matrix}{{{\eta(k)} = {{C_{2}\left\lbrack {S\;{\eta\left( \frac{\omega}{2\pi\; T} \right)}{{G(\omega)}}^{2}{{H_{a}\left( \frac{\omega}{2\pi\; T} \right)}}^{2}} \right\rbrack}*\left\lbrack \frac{\sin^{2}\left( {M\;\omega} \right)}{\sin^{2}\left( \frac{\omega}{2} \right)} \right\rbrack}}}_{\omega = \frac{\pi\; k}{M}} & (6)\end{matrix}$where C₂ is another constant and * denotes the convolution. Assumingthat the noise in the bands corresponding to unused tones issufficiently attenuated by the anti-alias filter, η(k) is approximatelyequal to:

$\begin{matrix}{{{\eta(k)} \approx {C_{3}{\sum\limits_{l = M_{1}}^{M_{2}}{{S_{\eta}\left( \frac{l}{2{MT}} \right)}{{G\left( \frac{\pi\; l}{M} \right)}}^{2}{{H_{a}\left( \frac{l}{2{MT}} \right)}}^{2}\left( {{\alpha\left( {k - l} \right)} + {\alpha\left( {{2M} - k - l} \right)}} \right)}}}},} & (7)\end{matrix}$where α(n) is defined as:

$\begin{matrix}{{{\alpha(n)} = {\int_{- \frac{\pi}{2M}}^{\frac{\pi}{2M}}{\left\lbrack \frac{\sin^{2}\left( {M\left( {\frac{\pi\; n}{M} - \lambda} \right)} \right)}{\sin^{2}\left( {\frac{1}{2}\left( {\frac{\pi\; n}{M} - \lambda} \right)} \right)} \right\rbrack{\mathbb{d}\lambda}}}},} & (8)\end{matrix}$M₁ . . . M₂ are the used tones and C₃ another constant. Defining x to bethe vector of frequency magnitudes to be solved for as:

$\begin{matrix}{{x\overset{\Delta}{=}\begin{bmatrix}{{G\left( \frac{\pi\; M_{1}}{M} \right)}}^{2} \\\vdots \\{{G\left( \frac{\pi\; M_{2}}{M} \right)}}^{2}\end{bmatrix}},} & (9)\end{matrix}$the SNR in bin k can be seen to be of the form

$\frac{b_{k}x_{k}}{a_{k}^{T}x},$in which b_(k) are scalars and a_(k) are vectors.

To determine the target spectral response, x is chosen to maximize thebit throughput. Approximating the capacity of bin k by √{square rootover (SNR(k))}, the optimal spectral profile is obtained by minimizingthe cost function F, where:

$\begin{matrix}{{F(x)} = {- {\sum\limits_{k = M_{1}}^{M_{2}}{\sqrt{\frac{b_{k}x_{k}}{a_{k}^{T}x}}.}}}} & (10)\end{matrix}$This minimization is performed over:X={xεR ^(N) :∥x∥=1, x _(i)≧0, 1≦i≦N}  (11)and can be accomplished by any one of a variety of constrainedoptimization strategies, as discussed in Bertsekas, NonlinearProgramming, Athena Scientific, Belmont, Mass., 1995. A median filtermay be applied to the output of the optimization algorithm to smooth theresulting target spectral response.

Let

$A\overset{\Delta}{=}{{H_{({0:{m_{h} + m_{g} - 2}})}^{m_{g}}\mspace{25mu}{and}\mspace{20mu} B}\overset{\Delta}{=}{H_{({m:{m + m_{l} - 1}})}^{m_{g}}.}}$The energy constraint in equation (3) can be written as:

$\begin{matrix}{S^{m} = {\left\{ {g \in {{R^{m_{g}}\text{:}\frac{g^{T}B^{T}{Bg}}{g^{T}A^{T}{Ag}}} \geq \alpha}} \right\}.}} & (12)\end{matrix}$

Matrix A has full column rank, since it corresponds to a fullconvolution, so R_(A) ΔA^(T)A is invertible. Let R_(B) ΔB^(T)B. Definingq=R_(A) ^(0.5)h, where R_(A) ^(0.5) is the square root of the positivedefinite matrix R_(A), the energy constraint set can be written in termsof q as:

$\begin{matrix}{Q\overset{\Delta}{=}{\left\{ {q \in {{R^{m_{g}}\text{:}\frac{q^{T}R_{A}^{- 0.5}R_{B}R_{A}^{- 0.5}q}{q^{T}q}} \geq \alpha}} \right\}.}} & (13)\end{matrix}$

The next step is to reduce the dimensionality, i.e., the number ofvariables to search over in the optimization process. For example, in avideo digital subscriber line(VDSL) application, an impulse shorteningfilter having a few hundred taps may be required. Searching over avariable space so large is difficult and impractical. Instead, theoptimization is performed by searching over a cleverly chosenlower-dimensional subset of the variable space. This simplification inthe optimization process may be done without significant reduction inthe performance of the communication system.

The reduction in dimensionality is accomplished by a transformation ofvariables. Let CΔR_(A) ^(−0.5)R_(B)R_(A) ^(−0.5). Let S=UΣU^(T) be thesingular-value decomposition of C, where Σ is a diagonal matrix whose(i, i)th element is σ_(i) and where the σ_(i) are arranged in descendingorder. If σ₁<α, there is no feasible solution corresponding to delay m.If σ₁≧α, let

$\begin{matrix}{{U = {{\left\lbrack {U_{1}\mspace{20mu} U_{2}} \right\rbrack\text{:}\sum} = \begin{bmatrix}\sum_{1} & 0 \\0 & \sum_{2}\end{bmatrix}}},} & (14)\end{matrix}$where U₁ has size (m_(g), m_(d)) and Σ₁ has size (m_(d), m_(d)) for somem_(d). These equations define the dimension of the range space of thematrix U₁, over which the search is confined. The dimension m_(d) may bechosen either to include all σ_(i) greater than some threshold β,β<α, orm_(d) may be a fixed number. Simulations indicate that m_(d) can be lessthan m_(g) by more than an order of magnitude without significantlyaffecting communication system performance.

The dimensionality reduction is achieved by a further transformation ofvariables: q=U₁ν. The energy constraint set now becomes:

$\begin{matrix}{V\overset{\Delta}{=}{\left\{ {v \in {{R^{m_{d}}\text{:}\frac{v^{T}{\sum_{1}v}}{v^{T}v}} \geq \alpha}} \right\} = \left\{ {{v \in {{R^{m_{d}}\text{:}v} \neq 0}},{\alpha{{v\left. ^{2}{{{- v^{T}}{\sum_{1}v}} \leq 0} \right\}}}}} \right.}} & (15)\end{matrix}$using the identity U₁ ^(T)CU₁=Σ₁.

Next, the cost function is expressed in terms of v. For a particular v,the corresponding g is given by g=R_(A) ^(−0.5)U₁v, which leads to:

$\begin{matrix}{{\begin{bmatrix}{G\left( \omega_{1} \right)} \\\vdots \\{G\left( \omega_{N} \right)}\end{bmatrix} = F_{g}};{F = \begin{bmatrix}1 & {\mathbb{e}}^{{- j}\;\omega_{1}} & {\mathbb{e}}^{{- j}\; 2\omega_{1}} & \cdots & {\mathbb{e}}^{{- j}\;{({m_{g} - 1})}\omega_{1}} \\\vdots & \vdots & \vdots & \vdots & \vdots \\1 & {\mathbb{e}}^{{- j}\;\omega_{1}} & {\mathbb{e}}^{{- j}\;\omega_{N}} & \cdots & {\mathbb{e}}^{{- j}\;{({m_{g} - 1})}\omega_{N}}\end{bmatrix}}} & (16)\end{matrix}$Let DΔFAR_(A) ^(−0.5)U₁. Let D^(R) Δreal (D) and D¹ Δ imag(D). D^(R) andD¹ are real matrices of size (N,m_(d)). Let d_(R,n) ^(T), 1≦n≦N andd_(1,n) ^(T),1≦n≦N be the rows of D^(R) and D¹, respectively. Then:

$\begin{matrix}{{{G\left( \omega_{n} \right)}} = \sqrt{\left( {d_{R,n}^{T}v} \right)^{2} + \left( {d_{1,n}^{T}v} \right)^{2}}} \\{= {\sqrt{{v^{T}\left( {{d_{R,n}d_{R,n}^{T}} + {d_{1,n}d_{1,n}^{T}}} \right)}v} = \sqrt{v^{T}T_{n}v}}}\end{matrix}$where Γ_(n) Δd_(R,n)d_(R,n) ^(T)+d_(1,n)d_(l,n) ^(T). These definitionsresult in:

$\begin{matrix}{{J_{o}(v)}\overset{\Delta}{=}{{j\left( {g(v)} \right)} = {\sum\limits_{n = 1}^{N}\left( {\sqrt{v^{T}\Gamma_{n}v} - {G_{d}\left( \omega_{n} \right)}} \right)^{2}}}} & (17)\end{matrix}$The gradient and Hessian of J_(o) are:

$\begin{matrix}{{\nabla_{J_{o}}(v)} = {\sum\limits_{n = 1}^{N}{2\left( {1 - \frac{G_{d}\left( \omega_{n} \right)}{\sqrt{v^{T}\Gamma_{n}v}}} \right)\;\Gamma_{n}v}}} & (18) \\{{H_{J_{o}}(v)} = {\sum\limits_{n = 1}^{N}{2\left\lbrack {{\left( {1 - \frac{G_{d}\left( \omega_{n} \right)}{\sqrt{v^{T}\Gamma_{n}v}}} \right)\;\Gamma_{n}} + {\frac{G_{d}\left( \omega_{n} \right)}{\left( \sqrt{v^{T}\Gamma_{n}v} \right)^{3}}\Gamma_{n}v\; v^{T}\Gamma_{n}}} \right\rbrack}}} & (19)\end{matrix}$

The projection P_(ν)(y) of any yεR^(m)d on to V is defined to be

$\begin{matrix}{{P_{V}(y)}\overset{\Delta}{=}{\arg\;{\min\limits_{v \in V}{{y - v}}^{2}}}} & (20)\end{matrix}$

To develop an algorithm to optimize the cost function J_(o) over V, anexpression must be derived for P_(ν)(y). There is no closed-formexpression for this projection operator, however there is a veryefficient algorithm to compute it. If yεv, P_(ν)(y)=y. If not, theprojection onto V is the same as the projection on to its boundary,defined by:boundary (V)Δ{νεR ^(m) ^(d) : ν≠∥ν∥²−ν^(T)Σ₁ν−0}  (21)The latter may be accomplished using LaGrange multipliers as follows.Defining a modified cost function S as:S(ν)Δ∥y−ν∥²+μ(α∥ν∥²−ν^(T)Σ₁ν)  (22)The equation ∇_(S)(ν)=0 must be solved on the boundary of V. Thisreduces to the following simultaneous equations:[(1+μα)I−μΣ ₁]ν=0  (23)α∥ν∥²−ν^(T)Σ₁ν=0  (24)Solving for ν in terms of μ from (23) and substituting in (24), thefollowing equation is obtained for μ:

$\begin{matrix}{{y^{T}{diag}\mspace{11mu}\left( \frac{a - \sigma_{i}}{\left\lbrack {1 + {\mu\left( {a - \sigma_{i}} \right)}} \right\rbrack^{2}} \right)y} = 0} & (25)\end{matrix}$Equation (25) may be written as a polynomial equation of order 2m_(d) inμ. The polynomial equation must be solved for the real and positiveroots, which may be done using one of the variety of efficientroot-finding algorithms in the literature. One of the real, positiveroots must be chosen by enumeration so that, when substituted into theexpression for ν, it leads to the smallest ∥y−ν∥².

Since the gradient and Hessian of the cost function J_(o) and theprojection operator onto V are available, the optimization can be donevery efficiently. For example, the penalty method may be used, asdescribed in Bertsekas, Nonlinear Programming, Athena Scientific,Belmont, Mass., 1995, or an iterative strategy consisting of gradientdescent followed by projection onto V. Both techniques have been testedand perform well.

Other embodiments are within the scope of the following claims.

1. A method for selecting an impulse response for a spectrallyconstrained impulse shortening filter in a multiple carriercommunication system, the method comprising: measuring received noisepower spectral density; computing a cost function using the noise power,the cost function being dependent on the impulse response; reducing thedimensionality of a space over which the cost function is defined; andminimizing the cost function.
 2. The method of claim 1, wherein thecommunication system includes a discrete Fourier transform and the noisepower spectral density is measured at an output of the discrete Fouriertransform.
 3. The method of claim 1, wherein the cost function is usedto compute coefficients for the spectrally constrained impulseshortening filter.
 4. Software on a processor readable medium comprisinginstructions for causing a processor in a communication system toperform the following operations: measure received noise power spectraldensity; compute a cost function using the noise power, the costfunction being dependent on an impulse response of a spectrallyconstrained impulse shortening filter; reduce the dimensionality of aspace over which the cost function is defined; and minimize the costfunction.
 5. The software of claim 4, wherein the cost function is usedto compute coefficients for the filter.
 6. A method for equalizing achannel in a multiple carrier communication system, the channel beingconfigured to receive a signal and including a spectrally constrainedimpulse shortening filter, the method comprising: measuring receivednoise power spectral density; computing a target spectral responsehaving a magnitude constraint that is based on the measured noise powerspectral density; selecting a frequency response of the spectrallyconstrained impulse shortening filter based on the target spectralresponse; and filtering the communication signal with the spectrallyconstrained impulse shortening filter.
 7. An article comprising amachine-readable medium storing machine-readable instructions that whenapplied to the machine cause the machine to: measure received noisepower spectral density; compute a target spectral response having amagnitude constraint that is based on the measured noise power spectraldensity; select a frequency response of a spectrally constrained impulseshortening filter based on the target spectral response; and filter acommunication signal with the spectrally constrained impulse shorteningfilter.
 8. An article comprising a machine-readable medium storingmachine-readable instructions that when applied to the machine cause themachine to: compute a target spectral response having a magnitudeconstraint that is based on measured noise power spectral density;shorten an impulse response of a channel in a multiple carriercommunication system so that a significant part of an energy of theimpulse response is confined to a region that is shorter than a targetlength; and filter a signal having a cyclic prefix based on the targetspectral response.